Cryptography Reference
In-Depth Information
equations:
minfP(A)g = 0
maxfP(A)g = A
P(A) = A=2
minfP(A
1
;A
2
)g = minfA
1
;A
2
g
maxfP(A
1
;A
2
)g = maxfA
1
;A
2
g
P(A
1
;A
2
) = A
1
=2 + A
2
=2
(11.2)
According to
Figure 11.3,
it is easy to verify the following equations:
A
W;a1
= P(A
1
) + P(A
2
) + P(A
3
) + A
4
A
B;a1
= P(A
1
) + P(A
2
) + P(A
3
) + A
0
1
+ A
0
2
+ A
0
3
+ A
0
4
A
W;a2
= P(A
0
1
;A
0
2
) + A
0
4
A
B;a2
= A
1
+ A
2
+ A
3
+ A
4
+ P(A
0
1
;A
0
2
) + A
0
3
A
W;b1
= P(A
0
1
+ A
0
2
) + A
0
3
A
B;b1
= A
1
+ A
2
+ A
3
+ A
4
+ P(A
0
1
+ A
0
2
) + A
0
4
A
W;b2
= P(A
1
) + P(A
2
) + P(A
3
)
A
B;b2
= P(A
1
) + P(A
2
) + P(A
3
) + A
4
+ A
0
1
+ A
0
2
+ A
0
3
+ A
0
4
(11.3)
The following Theorem 2 shows that the stacked shares with slight mis-
alignment can still recover the secret image. Note that, by saying the secret
image is recovered by its original color, we mean that a black (resp. white)
pixel in the secret image is represented by a black (resp. white) pixel in the
recovered secret image; and by saying the secret image is recovered by its
complementary color, we mean that a black (resp. white) pixel in the secret
image is represented by a white (resp. black) pixel in the recovered secret im-
age. In order to consist with Definition 1 and Definition 2, we generalize the
denition of contrast and average contrast as follows:
h l
2s
2
=
hl
2s
2
and =
(11.4)
where 2s
2
is the area of a pixel (including two subpixels) and the definitions
of h, l, h, and l are the same as that in Denition 1 and Denition 2.
Theorem 2 For a misaligned (2; 2)-VCS, denote (d
x
;d
y
) as the deviation of
the stacked shares, then the secret image can still be recovered if (d
x
;d
y
) falls
misaligned scheme are shown in
Table 11.3.
Proof: Denote l
blackarea
(resp. h
blackarea
) as the maximum (resp. minimum)
black area of a white (resp. black) pixel, denote l
blackarea
(resp. h
blackarea
) as
the average black area of a white (resp. black) pixel, denote h
whitearea
(resp.
l
whitearea
) as the minimum (resp. maximum) white area of a black (resp.
white) pixel, denote h
whitearea
(resp. l
whitearea
) as the average white area of
a black (resp. white) pixel. According to Figure 11.3, we have:
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